FU Berlin, Fachbereich Mathematik und Informatik, Institut für Informatik

Vortrag des Informatik-Kolloquiums

Abstract:  S-patches are multisided generalizations of rational Bezier surfaces. S-patches were first introduced over a dozen years ago by Loop and
DeRose, using barycentric coordinates for convex polygons and arrays of control points indexed by the vertices of high dimensional simplices.  Loop
and DeRose showed that S-patches are affine invariant, lie in the convex hull of their control points, and their boundaries are the Bezier curves determined by their boundary control points. S-patches also have an evaluation algorithm that is a straightforward generalization of the standard de Casteljau algorithm for Bezier triangles.

In this talk we begin by reviewing the construction and the basic properties and algorithms associated with standard three and four sided Bezier surfaces.  We go on to provide a general paradigm for building rational multisided generalizations of Bezier patches based on the notions of discrete convolution and Minkowski sum. We then show how to extend the S-patch construction of Loop and DeRose to arrays of control points indexed by arbitrary sets of p-tuples.

Like the S-patches of Loop and DeRose, these rational multisided Bezier patches are affine invariant, lie in the convex hull of their control points, and
their boundaries are the Bezier curves determined by their boundary control points.  They also have algorithms for evaluation, differentiation, and blossoming that generalize the corresponding algorithms for standard three sided and four sided Bezier schemes.  We end with a brief discussion of the advantages and disadvantages of multisided S-patches and their generalizations, along with a few hints about further extensions of multisided Bezier patches to toric Bezier schemes.